Question

Due to the current COVID pandemic conditions, assume that positive or negative status of any individual are equally likely. There are 3 members in a family. If one of the members has tested COVID positive, the conditional probability that at least 2 members are COVID positive is ______ (rounded off to three decimal places).

Hint:

Always remove outcomes that violate the given condition before applying conditional probability.

Answer 1

Correct Answer: 0.57

Let \(X\) = number of COVID positive members. Each member independently has probability \(0.5\) of being positive. Total possible cases for 3 members: \[ P(X=k) = \binom{3}{k}(0.5)^3 \] We are given: \[ \text{One member has tested COVID positive} \Rightarrow X \ge 1 \] We want: \[ P(X \ge 2 \mid X \ge 1) \] Compute probabilities: \[ P(X=0) = \frac{1}{8},\quad P(X=1) = \frac{3}{8},\quad P(X=2) = \frac{3}{8},\quad P(X=3) = \frac{1}{8} \] Thus: \[ P(X \ge 2) = \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = 0.5 \] \[ P(X \ge 1) = 1 - P(X=0) = 1 - \frac{1}{8} = \frac{7}{8} \] \[ P(X \ge 2 \mid X \ge 1) = \frac{0.5}{7/8} = \frac{4}{7} = 0.571 \] Thus the answer lies in: \[ \boxed{0.570\ \text{to}\ 0.572} \] Final Answer: 0.571